⇦Chinese Physics Letters, 2017, Vol. 34, No. 4, Article code 048102 Kinetics of Spherical Interface in Crystal Growth * Mei-Qin Fu(付美芹), Qing-Ling Bi(毕庆玲), Yong-Jun Lü(吕勇军)** Affiliations School of Physics, Beijing Institute of Technology, Beijing 100081 Received 11 January 2017 *Supported by the National Natural Science Foundation of China under Grant No 51171027.
**Corresponding author. Email: yongjunlv@bit.edu.cn Citation Text: Fu M Q, Bi Q L and Lü Y J 2017 Chin. Phys. Lett. 34 048102 Abstract The growth kinetics of spherical NiAl and CuZr crystals are studied by using molecular dynamics simulations. The growth rates of crystals are found to increase with the grain radius. The simulations show that the interface thickness and the Jackson $\alpha$-factor increase as the growth proceeds, indicating that the interface becomes increasingly rough during growth. Due to the increasing interface roughening, the fraction of repeatable growth sites at interface $f$ is proposed to actually increase in growth. An attachment rate, which is defined as the fraction of atoms that join the crystal interface without leaving, is used to approximate $f$, displaying a linear increase. With this approximation, we predict the growth rates as a function of the crystal radius, and the results qualitatively agree with those from the direct simulations. DOI:10.1088/0256-307X/34/4/048102 PACS:81.10.-h, 64.70.D-, 61.25.Mv © 2017 Chinese Physics Society Article Text The crystal growth in liquid metals is microscopically regarded as the advance of solid–liquid interface in liquids, and this process is controlled by the kinetics of atom attachment to the interface, the capillarity effect and the diffusion of thermal and mass near interface.[1] The growth rate of a crystal in melts depends upon the difference between the attachment and detachment rates of atoms at the solid–liquid interface. The attachment and the detachment of atoms are determined by the forward and back diffusion process in the liquid ahead of interface. In addition, the enthalpy increased with transferring from solid to liquid, and the entropy decreased in the reverse process are also related to the attachment and detachment. Hence, the temperature dependence of the growth rate[2] is written as $$\begin{alignat}{1} V=\frac{6aDf}{\sigma ^2}\Big[\exp\Big(-\frac{L}{kT_{\rm m}}\Big)-\exp\Big(-\frac{L}{kT}\Big)\Big],~~ \tag {1} \end{alignat} $$ where $a$ is the advancing distance of interface by adding an atom to the interface, $D$ is the diffusion coefficient in liquids, $T_{\rm m}$ is the equilibrium melting point, $\sigma$ is the atom diameter at interface, $f$ is the fraction of repeatable growth sites at interface, $L$ is the latent heat of crystallization, and $k$ is the Boltzmann constant. This growth mechanism reasonably predicts the temperature dependence of the growth rate in some semiconductors, intermetallic compounds and even pure metals that have low growth rate and high undercoolability.[3-6] In the growth mode, the $f$-factor depends on the interface roughness. For rough interfaces, $f$ is approximately 0.25 and shows independence of undercooling.[7] As the crystal grows, the interface becomes unstable under the influence of mass and heat transfer ahead of interface, eventually evolving into a dendritic form. Here we focus on the free and equiaxed growth, which commonly occurs in the initial period following the formation of critical nuclei. In this case, the effect of the curvature evolution on the interface kinetics has not been paid much attention in previous studies, and its mechanism is worthy of further investigation. In this Letter, we study the interface kinetics of spherical NiAl and CuZr crystals during the crystal growth using molecular dynamics (MD) simulations. The interfacial morphology, dynamic properties as well as their curvature dependence are discussed. MD method is used to simulate the equiaxed growth of NiAl and CuZr systems. Firstly, bulk crystals are prepared by arranging atoms based on the experimental crystal structures and lattice parameters. For NiAl and CuZr, the melts are solidified into single phase intermetallic compounds NiAl and CuZr with B2 structure. These crystals are then relaxed for 2 ns at room temperature and zero pressure. We fix the atoms in a spherical region with radius $r$. Then, the system is relaxed at high temperature, about 200 K above the experimental equilibrium melting point, for 2 ns to melt the remaining region into liquid, forming a solid–liquid coexistent system. By equilibrating at different temperatures, the solid–liquid coexistent temperature for the spherical crystal is determined when the solid–liquid interface maintains stable. The location of the solid–liquid interface is determined by using the structural order parameter $Q_{6}$.[8,9] The crystal growth is carried out at the temperature below the coexistent temperature. We use the embedded atom method (EAM) potentials to implement MD simulations of NiAl[10] and CuZr.[11] All simulations are performed in the isothermal–isobaric (NPT) ensemble with the systems size of $N=16000$ for CuZr and 54000 for NiAl. CuZr and NiAl alloys with the compositions of 50 at.% Zr and 50 at.% Al are prepared, respectively. All MD simulations are carried out using the LAMMPS code package.[12] We first calculate the equilibrium melting points of NiAl and CuZr. The liquid–solid–liquid tri-layer structures are relaxed at different trial temperatures. The temperature at which the planar solid–liquid interfaces keep equilibrium is approximate to the melting point. The calculations (1820 K for NiAl and 1299 K for CuZr) show the comparable results with the experimental measurements (1910 K for NiAl and 1208 K for CuZr[13]). The crystal growth is simulated from the spherical nucleus composed of 1000–1500 atoms. Firstly, the solid–liquid coexistent temperatures of these spherical nuclei are determined by relaxing the systems at different trial temperatures. Considering the slow growth of NiAl and CuZr, to achieve a distinct advance of the interface in a finite simulation time, the growth temperatures of NiAl and CuZr are set to 1600 K and 1100 K, 52 K and 75 K smaller than their coexistent temperatures, respectively. The homogeneous nucleation ahead of the interface never occurs in any of the simulations. Furthermore, the growth temperatures are far above the glass transition temperatures. Figure 1 shows the mean growth rate as a function of the crystal radius. The most remarkable finding is that the growth is accelerated with increasing the crystal radius in all the systems. In previous MD simulations, we have indicated that the crystal growth of NiAl and CuZr slabs is well described by Eq. (1).[14] If the spherical interface can be approximate to the planar interface under the condition of infinite radius, the crystal growth initiated from the critical nucleus actually undergoes an accelerating stage before the stable growth is reached.
cpl-34-4-048102-fig1.png
Fig. 1. Growth rates as a function of the crystal radius. (a) NiAl at $T=1600$ K, and (b) CuZr at $T=1100$ K. The solid lines are the predictions based on the crystal growth models.
Generally, the growth rate is determined by the solute distribution, thermal diffusion, curvature and kinetic effect. Because CuZr and NiAl are intermetallic compounds, the solute re-distribution can be excluded. In our simulations, constant temperature is imposed throughout the system, and the latent heat released during growth is assumed to transfer immediately. This assumption is reasonable in the initial stage of equiaxed growth due to weak heat releasing in undercooled melts. Then, we focus on the curvature effect and the interface kinetics. The curvature can depress the equilibrium temperature between liquid and solid, resulting in a curvature undercooling that is related to the interface energy by $\Delta T={2\gamma }/{r\Delta s}$, where $\gamma$ is the solid–liquid interface energy, and $\Delta s$ is the entropy of fusion per unit volume. Here $\gamma$ and $\Delta s$ are independent of the crystal size. Therefore, the curvature undercooling should decrease as the crystal grows, which does not contribute to the enhanced crystal growth. Excluding these aspects, it is naturally inferred that the underlying mechanism of the enhanced growth may be closely related to the interface kinetics.
cpl-34-4-048102-fig2.png
Fig. 2. Growth rates as a function of the interface thickness. (a) NiAl at $T=1600$ K, and (b) CuZr at $T=1100$ K. The solid lines are fits to linear relations.
The advance of the interface for a pure substance can be treated as a net rate at which atoms attach to and detach from the crystal interface. According to the crystal growth models presented in Eq. (1), the growth rate is given by the net rate multiplied by the thickness of a crystalline layer $a$ and the fraction of repeatable growth sites at the crystal interface $f$. The parameter $a$ is a constant at given temperature and pressure, and $f$ depends on the interface morphology. In general, $f$ is empirically set to 0.25 for rough interfaces, independent of undercooling. It is actually based on an underlying assumption that the interface roughness remains constant during growth. For the equiaxed growth of spherical crystals, the curvature radius continuously increases. It is necessary to examine the interface morphology during growth. We assume that the solid–liquid interface is a transition region between liquid and crystal rather than an atomic monolayer. The interface thickness can be obtained by fitting $Q_6(z)$ profile to a hyperbolic tangent function. Then, the simulation cell is divided into three regions: liquid, crystal and interface regions. We find that the interface thickness linearly increases with the growth rate, as shown in Fig. 2. The widening of the interface region reflects an increase of the interface roughness. Jackson proposed the $\alpha$-factor to measure the interface roughness at the atomic level[7,15,16] $$\begin{align} \alpha =\frac{L}{kT_{\rm m}}\frac{\eta }{Z},~~ \tag {2} \end{align} $$ where $\eta$ is the bond number at the interface, and $Z$ is the coordination number of crystal. If $\alpha>2$, the interface is smooth and the crystal growth is faceted; otherwise it is rough and the growth is non-faceted. In the definition of the Jackson $\alpha$-fact, the interface is regarded as a monolayer. In this work, we take the interface as a region with some thickness. Then, $\eta$ is the bond number in the interface region. If the interface becomes rougher during growth, the $\alpha$-factor is expected to increase. Figures 3(a) and 3(b) show that the $\alpha$-factor increases as the crystals grow, and the radius dependence is analogous to that of the interface thickness. It is noted that the values of $\alpha$-factor for the two alloys are less than 2, implying a tendency to non-faceted growth for NiAl and CuZr intermetallic compounds under present undercooling conditions. The $\alpha$-factor and the interface thickness consistently verify that the interface becomes rougher. It is further suggested that the $f$-factor is not constant for curved interfaces but depends on the curvature radius. An increasingly rough interface provides more repeatable sites during growth. The repeatable sites are those that are favorable for the attachment of atoms from the melt and against the detachment of atoms into the melt. In the crystal growth, the stably attached atoms without leaving are assumed to occupy the repeatable step sites. The attachment rate $f_{\rm a}$ is then defined as a ratio between the number of stably attached atoms $N_{\rm a}$ and that of atoms in the interface region $N_{\rm I}$. Correspondingly, the detachment rate $f_{\rm d}$ is defined as the ratio between the number of detached atoms and $N_{\rm I}$. Figures 3(c) and 3(d) show $f_{\rm a}$ as a function of radius for NiAl and CuZr corresponding to the cases in Fig. 1. It is clearly indicated that $f_{\rm a}$ increases with the radius. For NiAl, the growing $f_{\rm a}$ approaches to 0.25, which is close to the empirical value of $f$ for rough interface. For CuZr, the values are smaller than 0.20. The value of $f_{\rm a}$ is determined by the fraction of the stable growth site in the interface region, and the number of the latter is proportional to the interface area, $N_{\rm a} \propto 4\pi r^2$. However, due to the presence of the additional pressure induced by the curved interface, the value of $N_{\rm a}$ will further increase with radius. Therefore, we assume that $N_{\rm a}$ follows a power law $N_{\rm a} =Ar^\beta$, where the exponent $\beta$ should be larger than 2. A fit of $N_{\rm a}$ to the power law shows that the value of $\beta$ depends on the system, $\beta =2.6\pm 0.1$ for NiAl and $\beta =3.7\pm 0.2$ for CuZr. The number of atoms in an interface region $N_{\rm I}$ is related to the volume of interface region $N_I \propto 4\pi r^2\lambda$. Here $N_{\rm a}$ increases more rapidly than $N_{\rm I}$ with the radius, resulting in an increase of $f_{\rm a}$. In the size range of this study, $f_{\rm a}$ shows an approximately linear increase (Figs. 3(c) and 3(d)).
cpl-34-4-048102-fig3.png
Fig. 3. Jackson $\alpha$-factors as a function of the crystal radius for (a) NiAl and (b) CuZr. The solid curves are the exponential fits. Here (c) and (d) show the attachment rate of interface as a function of the crystal radius of NiAl and CuZr, respectively. The solid lines are linear fits.
We assume that the stable growth sites cover most repeatable step sites, and then use $f_{\rm a}$ to approximate the fraction of interface repeatable sites $f$. This approximation allows us to predict the growth rate by fitting $f_{\rm a}$. In Eq. (1), the prediction of the growth rates requires the knowledge of diffusion coefficients in liquids. For the equiaxed growth of spherical crystals, the effective diffusion process should refer to that normal to the curved interface, and the diffusion coefficient in Eq. (1) should be the normal diffusion coefficient near the interface. To calculate the near-interface $D_{\rm N}$, the average residence time of atoms in the interface region is first estimated, and then the mean-squared displacements (MSD) of the atoms in the interface region and in the average residence time are calculated in the spherical coordinates. From the time evolution of MSDs, we obtain $D_{\rm N}=7.43\times 10^{-5}\,{\rm cm}^2{\rm s}^{-1}$ for NiAl at $T=1600$ K and $D_{\rm N} =5.37\times 10^{-6}\,{\rm cm}^2{\rm s}^{-1}$ for CuZr at $T=1100$ K. The values of the lattice spacing corresponding to the low-index (100) plane are used in the calculation of NiAl and CuZr. Figure 1 shows that the estimated growth rates (solid lines) are comparable to the simulated results. These agreements indicate that the modification to the $f$-factor indeed accounts for the size dependence of the crystal growth. Due to the approximation of the linear relation between $f_{\rm a}$ and $r$, the growth rate is predicted to increase linearly. However, the growth rate tends to be slowed down as the growth proceeds, as shown in Fig. 1. We need a more accurate model to reveal the crystal-size dependence of the $f$-factor in further studies. In summary, we have studied the crystal growth of spherical NiAl and CuZr crystals by using MD simulations. Their growth rates are found to increase with the radius of crystals. The increase of the growth rate is interpreted as a result of the crystal-size dependence of the $f$-factor. An attachment rate, which is defined as the fraction of stable atoms at the crystal interface without leaving, is used to approximate the $f$-factor. By fitting the attachment rates obtained in simulations, the growth rates as a function of the crystal radius are predicted. The results qualitatively agree with the values from the direct simulations. References Molecular Dynamics Studies of Silicon Solidification and MeltingCrystal growth velocity in deeply undercooled Ni–Si alloysDiffusion-controlled crystal growth in deeply undercooled melt on approaching the glass transitionAccurate determination of crystal structures based on averaged local bond order parametersBond-orientational order in liquids and glassesDevelopment of an interatomic potential for the Ni-Al systemDevelopment of suitable interatomic potentials for simulation of liquid and amorphous Cu–Zr alloysFast Parallel Algorithms for Short-Range Molecular DynamicsMechanism of abnormally slow crystal growth of CuZr alloyKinetic Monte Carlo simulations of the surface roughening of binary systemsComment on the α-factor of Jackson for crystal growth from solution
[1]Kurz W and Fisher D J 1989 Fundamentals of Solidification (Switzerland: Trans Tech Publications) p 34
[2]Frenkel J 1932 Phys. Z. Sowjet Union 1 498
[3]Ashkenazy Y and Averback R S 2010 Acta Mater. 585 24
[4] Grabow M H, Gilmer G H and Bakker A F 1988 Mat. Res. Soc. Symp. Proc. 141 349
[5] Lü Y J 2012 Philos. Mag. Lett. 92 56
[6] Wang Q et al 2011 Phys. Rev. B 83 014202
[7]Jackson K A 2002 Inter. Sci. 10 159
[8] Lechner W and Dellago C 2008 J. Chem. Phys. 129 114707
[9] Steinhardt P J, Nelson D R and Ronchetti M 1983 Phys. Rev. B 28 784
[10] Pun G P and Mishin Y 2009 Philos. Mag. 89 3245
[11] Mendelev M I, Kramer M J, Ott R T, Sordelet D J, Yagodin D and Popel P 2009 Philos. Mag. 89 967
[12] Plimpton S 1995 J. Comput. Phys. 117 1
[13]Baker H 1992 ASM Handbook: Alloy Phase Diagram (US: ASM International) vol 3 pp 310, 780
[14] Yan X Q and Lü Y J 2015 J. Chem. Phys. 143 164503
[15] Bentz D N, Betush W J and Jackson K A 2003 J. Cryst. Growth 250 162
[16] Elwenspoek M 1986 J. Cryst. Growth 78 353